Wan-Tong Li

Hopf bifurcation and global periodic solutions in a delayed predator-prey system

This paper is concerned with a delayed predator-prey system with same feedback delays of predator and prey species to their growth, respectively. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases monotonously from zero. By using the theory of normal norm and center manifold reduction, an explicit algorithm for determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is derived. In addition, the global existence results of periodic solutions bifurcating from Hopf bifurcations are established by using a global Hopf bifurcation result due to Wu [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799-4838]. Finally, a numerical example supporting our theoretical prediction is also given. Our findings are contrasted with recent studies on a delayed predator-prey system with the feedback time delay of prey species to its growth by Song and Wei [Y. Song, J. Wei, Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, J. Math. Anal. Appl. 301 (2005) 1-21]. As the feedback time delay τ increases monotonously from zero, the positive equilibrium of the latter switches k times from stability to instability to stability. In contrast, the positive equilibrium of our system appears to lose the above property.

Periodic solutions of delayed ratio-dependent predator-prey models with monotonic or nonmonotonic functional responses

Abstract In this paper, by using the continuation theorem of coincidence degree theory, we establish several existence results of positive periodic solutions for the delayed ratio-dependent predator–prey model x′(t)=x(t) a(t)−b(t) ∫ −∞ t K(t−s)x(s) d s −c(t)g x(t) y(t) y(t), y′(t)=y(t) e(t)g x(t−τ(t)) y(t−τ(t)) −d(t) , when functional response function g is a monotonic or nonmonotonic, where a ( t ), b ( t ), e ( t ), τ ( t ) and d ( t ) are all positive periodic continuous functions with period ω >0, K is a dense function. As corollaries, some applications are listed. In particular, our results extend some known criteria.

Entire Solutions in Delayed Lattice Differential Equations with Monostable Nonlinearity

We construct new types of entire solutions for a class of monostable delayed lattice differential equations with global interaction by mixing a heteroclinic orbit of the spatially averaged ordinary differential equations with traveling wave fronts with different speeds. We also establish the uniqueness of entire solutions and the continuous dependence of such an entire solution on parameters, such as wave speeds, for the spatially discrete Fisher-KPP equation.

Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure

This paper is concerned with the existence of traveling wave solutions of a delayed predator-prey system with stage structure and nonlocal diffusion. By introducing the partial quasi-monotone condition and cross-iteration scheme, we first consider a class of delayed systems with nonlocal diffusion and deduce the existence of traveling wave solutions to the existence of a pair of upper-lower solutions. When the result is applied to the predator-prey system, we establish the existence of traveling wave solutions, as well as its precisely asymptotic behavior. Our result implies that there is a transition zone moving from the steady state with no species to the steady state with the coexistence of both species.

Hopf bifurcation and stability of periodic solutions in a delayed eco-epidemiological system

Abstract In this paper, a delayed predator–prey epidemiological system with disease spreading in predator population is considered. By regarding the delay as the bifurcation parameter and analyzing the characteristic equation of the linearized system of the original system at the positive equilibrium, the local asymptotic stability of the positive equilibrium and the existence of local Hopf bifurcation of periodic solutions are investigated. Moreover, we also study the direction of Hopf bifurcations and the stability of bifurcated periodic solutions, an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, numerical simulations supporting the theoretical analysis are also included.

Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator–prey models

Abstract This paper is concerned with two-species spatial homogeneous and inhomogeneous predator–prey models with Beddington–DeAngelis functional response. For the spatial homogeneous model, the asymptotic behavior of the interior equilibrium and the existence of Hopf bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered. Furthermore, the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are investigated. For the model with no-flux boundary conditions, Turing instability of the interior equilibrium solution is studied. In particular, Turing instability region regarding the parameters is established. Finally, to verify our theoretical results, some numerical simulations are also included.

Multiple positive solutions for p-Laplacian m-point boundary value problems on time scales

Let T be a time scale such that 0,[emailxa0protected]?T, ai>=0 for i=1,...,m-2. Let @xi satisfy 0<@x1<@x2<...<@xm-2<@r(T) and @?i=1^m^-^2ai<1. We consider the following p-Laplacian m-point boundary value problem on time scales(@fp(u^@D(t)))^@?+a(t)f(t,u(t))=0,[emailxa0protected]?(0,T),u(0)=0,@fp(u^@D(T))[emailxa0protected]?i=1m-2a[emailxa0protected]p(u^@D(@xi)),where [emailxa0protected]?Cld((0,T),[0,~)) and [emailxa0protected]?Cld((0,T)x[0,~),[0,~)). Some new results are obtained for the existence of at least twin or triple positive solutions of the above problem by applying Avery-Henderson and Leggett-Williams fixed point theorems respectively. In particular, our criteria extend and improve some known results.

Positive solutions of singular p-Laplacian BVPs with sign changing nonlinearity on time scales

We investigate a class of singular m-point p-Laplacian boundary value problem on time scales with the sign changing nonlinearity. By using the well-known Schauder fixed point theorem and upper and lower solutions method, some new existence criteria for positive solutions of the boundary value problem are presented. These results are new even for the corresponding differential (T=R) and difference equations (T=Z), as well as general time scales setting. As an application, an example is given to illustrate the results.

Traveling waves for a nonlocal dispersal SIR model with delay and external supplies

This paper is concerned with the existence, nonexistence and minimal wave speed of traveling waves of a nonlocal dispersal delayed SIR model with constant external supplies and Holling-II incidence rate. We find that the existence and nonexistence of traveling waves of the system are not only determined by the minimal wave speed c ? , but also by the so-called basic reproduction number R 0 of the corresponding reaction system. That is, we establish the existence of traveling waves for R 0 1 and each wave speed c ? c ? , and the nonexistence for R 0 1 and any 0 < c < c ? or R 0 < 1 . We also discuss how the latency of infection and the spatial movement of the infective individuals affect the minimal wave speed. Biologically speaking, the longer the latency of infection in a vector is, the slower the disease spreads.

Existence of positive periodic solution of a neutral impulsive delay predator-prey system

Abstract Sufficient conditions are obtained for the existence of periodic positive solution of the neutral impulsive delay predator–prey system. This is the first time that positive solution for impulsive delay predator–prey system is obtained by using the theory of coincidence degree. Our results in this paper indicate that under the appropriate linear periodic impulsive perturbations, the neutral impulsive delay predator–prey system preserves the original periodicity of the neutral nonimpulsive delay predator–prey system.